Given a configuration $C$ of $N$ distinct fixed points of equal mass in the plane (eventually, in space), let $f_C(N)$ denote the number of points $P$ for which the gravitational field at $P$ vanishes.
For example $f_C(2)=1$ for all $C$ consisting of two points, and for a $C$ consisting of $3$ collinear points, $f_C(3)=3$.
I conjecture that $f_C(N)$ is always finite and nonzero.
Is this true?